   Chapter 11.10, Problem 24E

Chapter
Section
Textbook Problem

# Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence.24. f(x) = cos x, a = π/2

To determine

To find: The Taylor series for f(x) centered at π2 and radius of convergence.

Explanation

Result used:

(1) If f has a power series expansion at a , f(x)=n=0f(n)(a)n!(xa)n , f(x)=f(a)+f(a)1!(xa)+f(a)2!(xa)2+f(a)3!(xa)3+

(2) The Ratio Test:

(i) If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent (and therefore convergent.)

(ii) If limn|an+1an|=L>1 or limn|an+1an|= , then the series n=1an is divergent.

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an .

Calculation:

Consider the function f(x)=cosx centered at a=π2 .

Since the function f(x)=cosx at π2 is f(π2)=0 .

Find the first derivative of f(x) at a=π2 .

f(x)=ddx(cosx)=sinx

f(x)=sinx (1)

Obtain f(π2) .

f(π2)=sin(π2)=1

Find the second derivative of f(x) at a=π2 .

f(2)(x)=d2dx2(cosx)=ddx(f(x))     [ddx(cosx)=f(x)]=ddx(sinx)    (by equation(1))=cosx

That is, f(2)(x)=cosx (2)

Compute f(2)(π2) .

f(2)(π2)=cos(π2)=0

That is, f(2)(π2)=0

Find the third derivative of f(x) at a=π2 as follows.

f(3)(x)=d3dx3(cosx)=ddx(f(2)(x))=ddx(cosx)    (by equation(2))=(sinx)

That is, f(3)(x)=sinx (3)

Compute f(3)(π2) .

f(3)(π2)=sin(π2)=1

Find the fourth derivative of f(x) at a=π2 .

f(4)(x)=d4dx4(cosx)=ddx(f(3)(x))=ddx(sinx)    (by equation(3))=cosx

That is, f(4)(x)=cosx (4)

Compute f(4)(π2) .

f(4)(π2)=cos(π2)=0

Find the fifth derivative of f(x) at a=π2 .

f(5)(x)=d5dx5(cosx)=ddx(f(3)(x))=ddx(cosx)    (by equation(3))=sinx

That is, f(5)(x)=sinx (4)

Compute f(5)(π2)

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